Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 15

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}\\ \frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(t\_1, t\_1, 2\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ 2.0 (fma (/ 1.0 t) t t)))))
   (/ 1.0 (/ 1.0 (- 1.0 (/ 1.0 (fma t_1 t_1 2.0)))))))

C

double code(double t) {
	double t_1 = 2.0 - (2.0 / fma((1.0 / t), t, t));
	return 1.0 / (1.0 / (1.0 - (1.0 / fma(t_1, t_1, 2.0))));
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(2.0 / fma(Float64(1.0 / t), t, t)))
	return Float64(1.0 / Float64(1.0 / Float64(1.0 - Float64(1.0 / fma(t_1, t_1, 2.0)))))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(2.0 / N[(N[(1.0 / t), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(1.0 / N[(1.0 - N[(1.0 / N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;\frac{1}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (/ 1.0 (- 1.2 (/ (- -0.32 (/ (- 0.032 (/ 0.0768 t)) t)) t)))
   (-
    1.0
    (/ 1.0 (+ (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 1.0 / (1.2 - ((-0.32 - ((0.032 - (0.0768 / t)) / t)) / t));
	} else {
		tmp = 1.0 - (1.0 / (((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(1.0 / Float64(1.2 - Float64(Float64(-0.32 - Float64(Float64(0.032 - Float64(0.0768 / t)) / t)) / t)));
	else
		tmp = Float64(1.0 - Float64(1.0 / Float64(Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(1.0 / N[(1.2 - N[(N[(-0.32 - N[(N[(0.032 - N[(0.0768 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{6}{5} + \left(\frac{\frac{4}{125}}{{t}^{2}} + \frac{8}{25} \cdot \frac{1}{t}\right)\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \color{blue}{\left(\frac{8}{25} \cdot \frac{1}{t} + \frac{\frac{4}{125}}{{t}^{2}}\right)}\right) - \frac{48}{625} \cdot \frac{1}{{t}^{3}}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \frac{\frac{4}{125}}{{t}^{2}}\right)} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}} \]

      3. associate--l+ N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{{t}^{3}}\right)}} \]

      4. cube-mult N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot t\right)}}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \frac{1}{t \cdot \color{blue}{{t}^{2}}}\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\frac{\frac{4}{125}}{{t}^{2}} - \frac{48}{625} \cdot \color{blue}{\frac{\frac{1}{t}}{{t}^{2}}}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\frac{\frac{4}{125}}{{t}^{2}} - \color{blue}{\frac{\frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{{t}^{2}}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{\color{blue}{t \cdot t}}} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t}}} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}{t}\right)\right)\right)\right)}} \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(\frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}\right)}{t}}\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{1}{\left(\frac{6}{5} + \frac{8}{25} \cdot \frac{1}{t}\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{-1 \cdot \frac{\frac{4}{125} - \frac{48}{625} \cdot \frac{1}{t}}{t}}}{t}\right)\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot {t}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-*r* N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) + 4\right)} \cdot t\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\left(\color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) \cdot t} + 4\right) \cdot t\right) \cdot t} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\mathsf{fma}\left(t \cdot \left(12 + -16 \cdot t\right) - 8, t, 4\right)} \cdot t\right) \cdot t} \]

      9. sub-neg N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{t \cdot \left(12 + -16 \cdot t\right) + \left(\mathsf{neg}\left(8\right)\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\left(12 + -16 \cdot t\right) \cdot t} + \left(\mathsf{neg}\left(8\right)\right), t, 4\right) \cdot t\right) \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\left(12 + -16 \cdot t\right) \cdot t + \color{blue}{-8}, t, 4\right) \cdot t\right) \cdot t} \]

      12. lower-fma.f64 N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 + -16 \cdot t, t, -8\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      13. +-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-16 \cdot t + 12}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      14. lower-fma.f64 100.0

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-16, t, 12\right)}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

   5. Applied rewrites 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;\frac{1}{1.2 - \frac{-0.32 - \frac{0.032 - \frac{0.0768}{t}}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.04938271604938271}{\left(t \cdot t\right) \cdot t}\right) - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (-
    (+ 0.8333333333333334 (/ 0.04938271604938271 (* (* t t) t)))
    (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t))
   (-
    1.0
    (/ 1.0 (+ (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = (0.8333333333333334 + (0.04938271604938271 / ((t * t) * t))) - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	} else {
		tmp = 1.0 - (1.0 / (((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(Float64(0.8333333333333334 + Float64(0.04938271604938271 / Float64(Float64(t * t) * t))) - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	else
		tmp = Float64(1.0 - Float64(1.0 / Float64(Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(0.8333333333333334 + N[(0.04938271604938271 / N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 4.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   7. Applied rewrites 4.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{5}{6} + \color{blue}{\left(\frac{4}{81} \cdot \frac{1}{{t}^{3}} + \frac{\frac{1}{27}}{{t}^{2}}\right)}\right) + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \frac{\frac{1}{27}}{{t}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\frac{1}{27}}{{t}^{2}} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\frac{1}{27}}{\color{blue}{t \cdot t}} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\frac{\color{blue}{\frac{1}{27} \cdot 1}}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\color{blue}{\frac{1}{27} \cdot \frac{1}{t}}}{t} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \color{blue}{\left(\frac{\frac{1}{27} \cdot \frac{1}{t}}{t} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\frac{1}{27} \cdot \frac{1}{t}}{t} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \left(\frac{\frac{1}{27} \cdot \frac{1}{t}}{t} - \frac{\color{blue}{\frac{2}{9}}}{t}\right) \]

      12. div-sub N/A

          \[\leadsto \left(\frac{5}{6} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right) + \color{blue}{\frac{\frac{1}{27} \cdot \frac{1}{t} - \frac{2}{9}}{t}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\left(\frac{0.04938271604938271}{\left(t \cdot t\right) \cdot t} + 0.8333333333333334\right) - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot {t}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-*r* N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) + 4\right)} \cdot t\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\left(\color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) \cdot t} + 4\right) \cdot t\right) \cdot t} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\mathsf{fma}\left(t \cdot \left(12 + -16 \cdot t\right) - 8, t, 4\right)} \cdot t\right) \cdot t} \]

      9. sub-neg N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{t \cdot \left(12 + -16 \cdot t\right) + \left(\mathsf{neg}\left(8\right)\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\left(12 + -16 \cdot t\right) \cdot t} + \left(\mathsf{neg}\left(8\right)\right), t, 4\right) \cdot t\right) \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\left(12 + -16 \cdot t\right) \cdot t + \color{blue}{-8}, t, 4\right) \cdot t\right) \cdot t} \]

      12. lower-fma.f64 N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 + -16 \cdot t, t, -8\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      13. +-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-16 \cdot t + 12}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      14. lower-fma.f64 100.0

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-16, t, 12\right)}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

   5. Applied rewrites 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.04938271604938271}{\left(t \cdot t\right) \cdot t}\right) - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
     t))
   (-
    1.0
    (/ 1.0 (+ (* (* (fma (fma (fma -16.0 t 12.0) t -8.0) t 4.0) t) t) 2.0)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	} else {
		tmp = 1.0 - (1.0 / (((fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0));
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	else
		tmp = Float64(1.0 - Float64(1.0 / Float64(Float64(Float64(fma(fma(fma(-16.0, t, 12.0), t, -8.0), t, 4.0) * t) * t) + 2.0)));
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / N[(N[(N[(N[(N[(N[(-16.0 * t + 12.0), $MachinePrecision] * t + -8.0), $MachinePrecision] * t + 4.0), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot {t}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-*r* N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      4. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) + 4\right)} \cdot t\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\left(\color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right) - 8\right) \cdot t} + 4\right) \cdot t\right) \cdot t} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{\mathsf{fma}\left(t \cdot \left(12 + -16 \cdot t\right) - 8, t, 4\right)} \cdot t\right) \cdot t} \]

      9. sub-neg N/A

         \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{t \cdot \left(12 + -16 \cdot t\right) + \left(\mathsf{neg}\left(8\right)\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\left(12 + -16 \cdot t\right) \cdot t} + \left(\mathsf{neg}\left(8\right)\right), t, 4\right) \cdot t\right) \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\left(12 + -16 \cdot t\right) \cdot t + \color{blue}{-8}, t, 4\right) \cdot t\right) \cdot t} \]

      12. lower-fma.f64 N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(12 + -16 \cdot t, t, -8\right)}, t, 4\right) \cdot t\right) \cdot t} \]

      13. +-commutative N/A

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-16 \cdot t + 12}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

      14. lower-fma.f64 100.0

          \[\leadsto 1 - \frac{1}{2 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-16, t, 12\right)}, t, -8\right), t, 4\right) \cdot t\right) \cdot t} \]

   5. Applied rewrites 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-16, t, 12\right), t, -8\right), t, 4\right) \cdot t\right) \cdot t + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (-
    0.8333333333333334
    (/
     (-
      0.2222222222222222
      (/ (+ (/ 0.04938271604938271 t) 0.037037037037037035) t))
     t))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (((0.04938271604938271 / t) + 0.037037037037037035) / t)) / t);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(Float64(Float64(0.04938271604938271 / t) + 0.037037037037037035) / t)) / t));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + 0.037037037037037035), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right) \cdot {t}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + t \cdot \left(t - 2\right), {t}^{2}, \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \left(t - 2\right) + 1}, {t}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - 2\right) \cdot t} + 1, {t}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{\frac{0.04938271604938271}{t} + 0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(0.2222222222222222, t, -0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (-
    0.8333333333333334
    (/ (fma 0.2222222222222222 t -0.037037037037037035) (* t t)))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - (fma(0.2222222222222222, t, -0.037037037037037035) / (t * t));
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(0.8333333333333334 - Float64(fma(0.2222222222222222, t, -0.037037037037037035) / Float64(t * t)));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.8333333333333334 - N[(N[(0.2222222222222222 * t + -0.037037037037037035), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      3. remove-double-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \left(\frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right)\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \color{blue}{\left(\frac{5}{6} - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) + \frac{5}{6}\right) - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)\right)} - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right) \]

      7. associate-+r- N/A

         \[\leadsto \color{blue}{\frac{5}{6} + \left(\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right) - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{5}{6} + \left(\left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2}{9} \cdot 1}{t}}\right)\right) - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} + \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2}{9}}}{t}\right)\right) - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      10. distribute-neg-frac2 N/A

          \[\leadsto \frac{5}{6} + \left(\color{blue}{\frac{\frac{2}{9}}{\mathsf{neg}\left(t\right)}} - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{\color{blue}{-1 \cdot t}} - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{{t}^{2}}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \left(\mathsf{neg}\left(\frac{\frac{1}{27}}{\color{blue}{t \cdot t}}\right)\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \left(\mathsf{neg}\left(\frac{\frac{\color{blue}{\frac{1}{27} \cdot 1}}{t}}{t}\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{27} \cdot \frac{1}{t}}}{t}\right)\right)\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \color{blue}{\frac{\frac{1}{27} \cdot \frac{1}{t}}{\mathsf{neg}\left(t\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + \left(\frac{\frac{2}{9}}{-1 \cdot t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{-1 \cdot t}}\right) \]

      18. div-sub N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{-1 \cdot t}} \]

   7. Applied rewrites 99.3%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{\mathsf{fma}\left(0.2222222222222222, t, -0.037037037037037035\right)}{t \cdot t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right) \cdot {t}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + t \cdot \left(t - 2\right), {t}^{2}, \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \left(t - 2\right) + 1}, {t}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - 2\right) \cdot t} + 1, {t}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{\mathsf{fma}\left(0.2222222222222222, t, -0.037037037037037035\right)}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666))
   (fma (fma (- t 2.0) t 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	} else {
		tmp = fma(fma((t - 2.0), t, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	else
		tmp = fma(fma(Float64(t - 2.0), t, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - 2.0), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{1}{6} + \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{1}{6} + \frac{\color{blue}{\frac{2}{9}}}{t}\right) \]

      4. lower-/.f64 99.0

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222}{t}}\right) \]

   5. Applied rewrites 99.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right) \cdot {t}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + t \cdot \left(t - 2\right), {t}^{2}, \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \left(t - 2\right) + 1}, {t}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - 2\right) \cdot t} + 1, {t}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t - 2, t, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{t - 2}, t, 1\right), {t}^{2}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      9. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (- 1.0 (+ (/ 0.2222222222222222 t) 0.16666666666666666))
   (fma (fma t -2.0 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 1.0 - ((0.2222222222222222 / t) + 0.16666666666666666);
	} else {
		tmp = fma(fma(t, -2.0, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(1.0 - Float64(Float64(0.2222222222222222 / t) + 0.16666666666666666));
	else
		tmp = fma(fma(t, -2.0, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(1.0 - N[(N[(0.2222222222222222 / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(t * -2.0 + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{6} + \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \left(\frac{1}{6} + \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 1 - \left(\frac{1}{6} + \frac{\color{blue}{\frac{2}{9}}}{t}\right) \]

      4. lower-/.f64 99.0

         \[\leadsto 1 - \left(0.16666666666666666 + \color{blue}{\frac{0.2222222222222222}{t}}\right) \]

   5. Applied rewrites 99.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} \cdot \left(1 + -2 \cdot t\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot t\right) \cdot {t}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + -2 \cdot t, {t}^{2}, \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot t + 1}, {t}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot -2} + 1, {t}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t, -2, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (fma (fma t -2.0 1.0) (* t t) 0.5)))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = fma(fma(t, -2.0, 1.0), (t * t), 0.5);
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = fma(fma(t, -2.0, 1.0), Float64(t * t), 0.5);
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(t * -2.0 + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 99.0

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2} \cdot \left(1 + -2 \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} \cdot \left(1 + -2 \cdot t\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot t\right) \cdot {t}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + -2 \cdot t, {t}^{2}, \frac{1}{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot t + 1}, {t}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot -2} + 1, {t}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(t, -2, 1\right)}, {t}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), \color{blue}{t \cdot t}, \frac{1}{2}\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -2, 1\right), t \cdot t, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.5 - t \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0)) 0.05)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (- 1.0 (- 0.5 (* t t)))))

C

double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 - (0.5 - (t * t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((2.0d0 / t) / ((1.0d0 / t) + 1.0d0)) <= 0.05d0) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 1.0d0 - (0.5d0 - (t * t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 1.0 - (0.5 - (t * t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if ((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 1.0 - (0.5 - (t * t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0)) <= 0.05)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(1.0 - Float64(0.5 - Float64(t * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (((2.0 / t) / ((1.0 / t) + 1.0)) <= 0.05)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 1.0 - (0.5 - (t * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.5 - N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))) < 0.050000000000000003

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 99.0

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if 0.050000000000000003 < (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} + -1 \cdot {t}^{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} - {t}^{2}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} - {t}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto 1 - \left(\frac{1}{2} - \color{blue}{t \cdot t}\right) \]

      5. lower-*.f64 99.6

         \[\leadsto 1 - \left(0.5 - \color{blue}{t \cdot t}\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto 1 - \color{blue}{\left(0.5 - t \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.05:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.5 - t \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.005:\\ \;\;\;\;1 - \left(0.5 - t \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0))) 0.005)
   (- 1.0 (- 0.5 (* t t)))
   0.8333333333333334))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 0.005) {
		tmp = 1.0 - (0.5 - (t * t));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((2.0d0 - ((2.0d0 / t) / ((1.0d0 / t) + 1.0d0))) <= 0.005d0) then
        tmp = 1.0d0 - (0.5d0 - (t * t))
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 0.005) {
		tmp = 1.0 - (0.5 - (t * t));
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 0.005:
		tmp = 1.0 - (0.5 - (t * t))
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0))) <= 0.005)
		tmp = Float64(1.0 - Float64(0.5 - Float64(t * t)));
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 0.005)
		tmp = 1.0 - (0.5 - (t * t));
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(1.0 - N[(0.5 - N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} + -1 \cdot {t}^{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 1 - \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} - {t}^{2}\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{2} - {t}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto 1 - \left(\frac{1}{2} - \color{blue}{t \cdot t}\right) \]

      5. lower-*.f64 99.6

         \[\leadsto 1 - \left(0.5 - \color{blue}{t \cdot t}\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto 1 - \color{blue}{\left(0.5 - t \cdot t\right)} \]

   if 0.0050000000000000001 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.005:\\ \;\;\;\;1 - \left(0.5 - t \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{2}{t + 1}\\ 1 - \frac{1}{\mathsf{fma}\left(t\_1, t\_1, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ 2.0 (+ t 1.0))))) (- 1.0 (/ 1.0 (fma t_1 t_1 2.0)))))

C

double code(double t) {
	double t_1 = 2.0 - (2.0 / (t + 1.0));
	return 1.0 - (1.0 / fma(t_1, t_1, 2.0));
}

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(2.0 / Float64(t + 1.0)))
	return Float64(1.0 - Float64(1.0 / fma(t_1, t_1, 2.0)))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{1 \cdot 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \cdot \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}}}} \]

   3. remove-double-div 100.0

      \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)}} \]

   4. lift-fma.f64 N/A

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\color{blue}{\frac{1}{t} \cdot t + t}}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\color{blue}{\frac{1}{t}} \cdot t + t}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)} \]

   6. lft-mult-inverse N/A

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\color{blue}{1} + t}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)} \]

   7. +-commutative N/A

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\color{blue}{t + 1}}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)} \]

   8. lower-+.f64 100.0

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\color{blue}{t + 1}}, 2 - \frac{2}{\mathsf{fma}\left(\frac{1}{t}, t, t\right)}, 2\right)} \]

   9. lift-fma.f64 N/A

      \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{\color{blue}{\frac{1}{t} \cdot t + t}}, 2\right)} \]

   10. lift-/.f64 N/A

       \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{\color{blue}{\frac{1}{t}} \cdot t + t}, 2\right)} \]

   11. lft-mult-inverse N/A

       \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{\color{blue}{1} + t}, 2\right)} \]

   12. +-commutative N/A

       \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{\color{blue}{t + 1}}, 2\right)} \]

   13. lower-+.f64 100.0

       \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{\color{blue}{t + 1}}, 2\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{t + 1}, 2 - \frac{2}{t + 1}, 2\right)}} \]
7. Add Preprocessing

Alternative 13: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0))) 0.005)
   (fma t t 0.5)
   0.8333333333333334))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 0.005) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0))) <= 0.005)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 0.0050000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 0.0050000000000000001 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= (- 2.0 (/ (/ 2.0 t) (+ (/ 1.0 t) 1.0))) 1.0) 0.5 0.8333333333333334))

C

double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((2.0d0 - ((2.0d0 / t) / ((1.0d0 / t) + 1.0d0))) <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (Float64(2.0 - Float64(Float64(2.0 / t) / Float64(Float64(1.0 / t) + 1.0))) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((2.0 - ((2.0 / t) / ((1.0 / t) + 1.0))) <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], 0.5, 0.8333333333333334]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t)))) < 1

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1 < (-.f64 #s(literal 2 binary64) (/.f64 (/.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) t))))

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 - \frac{\frac{2}{t}}{\frac{1}{t} + 1} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.0% accurate, 101.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 59.6%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))